Interactive Mathematics For Engineers With MAPLE Usage - UTCB

INTERACTIVE MATHEMATICS FOR ENGINEERS WITH MAPLE1 USAGE Ralitsa Vasileva-Ivanova, Pr. Assist. Prof, PhD, University of Ruse, Bulgaria, [email protected], Magdalena Petkova, PhD, Bulgaria, [email protected], Emiliya Velikova, Assoc. Prof., PhD, University of Ruse, Bulgaria, [email protected], Ion Mierlus-Mazilu, Assoc. Prof., PhD, Technical University of Civil Engineering, Bucharest, Romania, [email protected]

Abstract: This article presents some possible usages of the interactive Maple Tutors in engineering mathematics, which correspond to the FutureMath project purposes. The Maple Tutors allow students to work step-by-step through mathematical problems. For example, one Maple Tutor provides students with opportunities to practice exercises by applying different rules of integration. They need to know the theoretical background, and after that have to understand what type of a problem it is necessary to solve (typical or complicated). They can also use the Maple Assistant to support their solution of the problem and to experiment with it. Students can perform some steps themselves, and ask Maple to perform the next ones. In addition, in this article are presented some mathematical problems from the National Student Olympiad in Computer Mathematic “Academician Stefan Dodunekov” (CompMath) in Bulgaria, solved and visualized by the usage of Maple 18. Keywords: Calculus, Maple Tutors, education, CompMath problems, FutureMath project

1. INTRODUCTION Nowadays, the usage of computer technologies for teaching, learning and research in mathematics is ever growing. The use of computer algebra systems (CAS) in education is still relatively low, but the growing body of research and the interest suggests that its extended use is imminent. There are many different types of CAS, including computer packages such as MATLAB, Maple, Mathematica, Wolfram Alpha, etc. and handheld calculators such as the NT-Nspire and Casio Classpad. The underlying concepts and proofs of many mathematical concepts involve difficult and abstract ideas that present a mountainous obstacle to many students. CAS offers both, an opportunity and a challenge to present novel approaches that assist students and teachers in the development of a better understanding of the concepts. They can be used for learning and teaching of complicated mathematical concepts and problem-solving and developing of higher-level cognitive skills (A. Kumar & S. Kumaresan, 2008, p. 374). Basic features of Maple, are described by I. Shingareva and C. Lizárraga-Celaya (2009, pp. 4-5). Some of them are: easy to use, help can be found within the problem on the Internet; fast symbolic, numerical computation, and interactive visualization; powerful programming language, intuitive syntax, easy debugging; two forms of interactive interface: a command-line and a graphical environment; free resources, collaborative character of development, Maple Application Centre, Teacher Resource Center, Student Help Center, Maple Community etc. Chii-Huei Yu (2016, p. 59) underlines that the rapid computations and the visually appealing graphical interface of Maple render creative research possibilities. The program is significant among mathematical calculation systems and can be considered a leading tool in the CAS field. The superiority of Maple lies in its simple instructions and ease of use, which enable beginners to learn the operating techniques in a short period. In addition, through the numerical and symbolic computations, performed by Maple, the process of thinking can be converted into a series of instructions.

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The Symbolic Computation Group at the University of Waterloo developed the first concept of Maple software and initial versions in the early 1980s.

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2. METHODS OF INSTRUCTIONS IN ENGINEERING MATH WITH MAPLE Maple is a math software that combines the world's most powerful math engine with an interface that makes it extremely easy to analyze, explore, visualize, and solve mathematical problems. It helps to solve mathematical problems easily and accurately; to solve math problems quickly; to solve problems from virtually any branch of mathematics or any field that relies on mathematics, such as calculus, algebra, differential equations, statistics, control design, linear algebra, physics, optimization, group theory, differential geometry, signal processing, special functions, number theory, financial modelling, etc.; and to gain insight into the problem, the solution, the data, or the concept by using a huge variety of customizable 2-D and 3-D plots and animations; to keep problems, solutions, visualizations, and explanations all together in a single, easy-to-follow document, so we don't have to waste time reconstructing our intellectual processes; to develop complex solutions using a sophisticated programming language designed for mathematics, so our code is shorter, easier to write, easier to debug, and easier to maintain; to create interactive applications for us, our students, or our colleagues, without having to be an expert programmer, and to also share them over the web (http://www.maplesoft.com, May 8, 2017), etc. Many researchers have used software Maple in math education, for example: 1) an effective learning strategy in integral calculus (T. Salleh & E. Zakaria, 2015, p.183); 2) a way to understand mathematics using Maple animations (A. Alharbi, F. Tcheir and M. Siddique, 2016, p. 148); 3) in the teaching of engineering mathematics (N. Kiyanovska, 2016), etc. In this paper, we are presenting some possible usages of the interactive Maple Tutors in engineering mathematics for university students – future engineers. The educational process is illustrated in Fig.1. Some of its components are: 1) The Maple Portal for Students acts as a starting point for hundreds of common tasks from mathematics’ courses and outruns the question "How do I ..." in Maple. It uses interactive assistants, context menus, palettes, and task templates. Answers over 130 “How do I ...” questions. Includes links to over a dozen new task templates. Provides information about additional resources. 2) Over 50 interactive Maple Tutors offer focused learning environments in which students can explore and reinforce fundamental mathematical concepts. Many of these Tutors allow students to work step-by-step through math problems. Students can perform steps themselves, ask for hints, or ask Maple to perform the next step. For example, one Maple Tutor applies the different rules of integration. Another provides help with performing Gaussian elimination on matrices, allowing students to focus on the steps without getting lost in the arithmetic. The Maple Tutors frequently use 2-D and 3-D plots and animations, reinforcing concepts that are sometimes difficult to visualize. Examples include volumes and surfaces of revolution, eigenvector plots, Newton’s method, gradients, space curves, conic sections, and the plots. Tutors are available for a single variable, multivariable, and vector calculus, precalculus, linear algebra, complex variables, numerical analysis, and differential equations. In calculus and engineering mathematics, there are many methods by which integral problems could be solved, for example, change of variables method, integration by parts method, partial fractions method, trigonometric substitution method, etc. 3) The ShowSolutions command would show all the steps for solving a problem, as well as record the rule or the method used for each step. The ShowSolutions command can be used for a single-variable differentiation, a limit, or an integration problem (indefinite integrals). 4) Some mathematical problems could not be solved using Maple Tutor or Maple Assistant. In that case, the students have to use the correct Maple commands. Therefore, they need to know the commands and their syntax. The commands that are involved in the solution of presented math problems, are described in Table 1. 2

START

THEORETICAL BACKGROUND NEEDED

USING MAPLE PORTAL FOR STUDENTS

YES

TYPICAL MATH PROBLEM?

NO

USING MAPLE COMMANDS

EXPERIMENTAL WORK

* > ShowSolution( );

TOOLS/TUTORS

USING MAPLE TUTORS

BUTTON LAUNCH

>Student[Calculus1][...Tutor]( );

END

Fig. 1. Graph of the educational process Table 1. List of Maple commands (Tonchev, 2013, pp. 232-235) SYNTAX > factor(a, K); a – expression K - field extension over which to factor > int(expression,x=a..b, options);

> diff(f, [x1$n]);

> solve(equations, variables); > fsolve( equations, variables, complex );

> pivot(A, i, j,); > dsolve(ODE); > dsolve({ODE, ICs}, y(x), options);

APPLICATION Computes the factorization of a multivariate polynomial with integer, rational, (complex) numeric, or algebraic number coefficients. Calling sequence computes the definite integral of the expression with respect to the variable x on the interval from a to b. Derivatives of nth order, where n is not specified as a number, can be constructed as in diff(f(x),[x$n]) and are interpreted as integer order derivatives, that is, computed assuming n is an integer. Solves one or more equations or inequalities for their unknowns. Numerically computes the zeroes of one or more equations, expressions or procedures. The pivot(a, i, j) function pivots a about the non-zero entry a[i, j]. Multiples of the ith row are added to every other row in a, with the result that all of the entries in the jth column of a are zero except for the (i, j)th element. As a general ode solver, dsolve handles different types of ode problems. Solving odes or a system of them with given initial conditions (boundary value problems).

> limit(f, x=a); f - algebraic expression a - algebraic expression; limit point, possibly infinity, or -infinity

This function attempts to compute the limiting value of f as x approaches a.

> extrema(expr, {constraints});

The function can be used to find extreme values of a multivariate expression with zero or more constraints.

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3. EXAMPLES The presented mathematical problems are a part of the National Student Olympiad in Computer Mathematics “Academician Stefan Dodunekov” (CompMath2, 2012-2016), Group B – Engineering and Natural Science. They are tested by using Maple 18 (2016). Problem 1 (CompMath 2016): Expanding irreducible polynomial into factors with real coefficients: x12  4 x11  5 x10  4 x9  4 x8  20 x 7  25 x 6  20 x5  x 4  16 x3  20 x 2  16 x  4 . Solution by using Maple Tutor: Ways of calling the Maple Tutor - from the Maple/Tools/Precalculus/Polynomials and Roots (Fig. 2) or by using the following commands, written in the Maple Worksheet: > with(Student[Precalculus]); > Student[Precalculus][PolynomialTutor]();

Fig. 2. Graphic of polynomial Solution by using standard commands: > with(Student[Calculus1]); > f := x^12+4*x^11+5*x^10+4*x^9-4*x^8-20*x^7-25*x^6-20*x^5x^4+16*x^3+20*x^2+16*x+4; > factor(f, real);

Problem 2 (CompMath 2012): Solve the following integral

 2

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 1  sin  x   cos  x dx

.

0

Step-by-step Solution of the indefinite integral: > with(Student[Calculus1]); > Int(1/(1+sin(x)+cos(x)), x) >

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CompMath is a mathematical contest for university students enrolled in a bachelor or master degree courses, which is organized once per year in Bulgaria (http://www.CompMath.eu/). 4

Solution by using Maple Tutor: Ways of calling the Maple Tutor - from the Maple/Tools/Tutors/Calculus1-Single Variable/Integration Methods (Fig. 3, 4) or by using the following commands, written in the Maple Worksheet: > with(Student[Calculus1]); > Student[Calculus1][IntTutor]();

Fig. 3. Enter a function. Make a change of variables

Fig. 4. All steps of the solution

Solution by using standard commands: > with(Student[Calculus1]); > int(1/(1+sin(x)+cos(x)), x = 0 .. (1/2)*Pi); Problem 3 (CompMath 2016): Solve the equation and find the extrema of the function 2  x  x  sin  x  . 3

Solution by using Maple Tutor: Ways of calling the Maple Tutor - from the Maple/Tools/Tutors Calculus1-Single Variable/Curve Analysis (Fig. 5÷7) or by using the following commands, written in the Maple Worksheet: > with(Student[Calculus1]); > Student[Calculus1][CurveAnalysisTutor]();

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Fig. 5. Find the root of the function

Fig. 6. Find the maximum of the function

Fig. 7. Find the minimum of the function

Solution by using standard commands: > with(Student[Calculus1]); > h := x-> 2+x^3-x-sin(x); > fsolve(h(x), x); > df := diff(h(x), x); > fsolve(df, x); > extrema(2+x^3-x-sin(x), {x});

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Problem 4 (CompMath 2012): Solve and analyse the system solution according to the ax1  x2  x3  1 parameter a values . x1  ax2  x3  1

x1  x2  ax3  a 2 Solution by using Maple Tutor: A way to call the Maple Tutor – Maple/Tools/Tutors/Linear Algebra/Linear System Solving (Fig. 8÷11).

Fig. 8. Enter the elements of the matrix

Fig. 9. Swap row 1 with row 2

Fig. 10. The linear system of equations

Fig. 11. The result

Solution by using standard commands: > with(LinearAlgebra); > A := Matrix(3, 4, {(1, 1) = a, (1, 2) = 1, (1, 3) = 1, (1, 4) = 1, (2, 1) = 1, (2, 2) = a, (2, 3) = 1, (2, 4) = 1, (3, 1) = 1, (3, 2) = 1, (3, 3) = a, (3, 4) = a^2}) > A := Pivot(A, 1, 1);

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> A := Pivot(A, 2, 2);

> A := Pivot(A, 3, 3);

> solve({x1+x2+a*x3 = a^2, x1+a*x2+x3 = 1, a*x1+x2+x3 = 1}, {x1, x2, x3});

Conclusions: 1) x1 is equal to x2; 2) the system linear equations is undefined if a  2 (it has countless solutions). Problem 5 (CompMath 2013): Draw the graphic of the function y(x), where y(x) is the solution of the Ordinary Differential Equation (ODE) y''  x   2 y  x   1 , and conditions y(0)=1 and y`(0)=0. Solution by using ODE Analyzer Assistant3: A way to call the Maple Tutor – from the Maple/Tools/Assistants/ODE Analyzer Assistant (Fig. 12 ÷15).

Fig. 12. Add the equation

Fig. 13. Add the conditions

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The ODE Analyzer Assistant is a point-and-click interface to the ODE solver routines. Using the assistant, you can compute numeric and exact solutions and plot the solutions. For more information, see dsolve[interactive] and worksheet/interactive/dsolve.

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Fig. 14. Solve the problem

Fig. 15. Plot the graphic of the solution

Solution by using standard commands: > with(Student[Calculus1]); > ode := diff(y(x), `$`(x, 2)) = 2*y(x)+1;

> dsolve(ode); > ics := y(0) = 1, (D(y))(0) = 0; > dsolve({ics, ode});

Problem 6 (CompMath 2012): Find the limit lim

 x 2

1  cot x tan x

.

Solution by using Maple Tutor: A way of calling the Maple Tutor – from the Maple/Tools/Calculus1-Single Variable/Limit Methods (Fig. 16 ÷ 18) or by using the following commands, written in the Maple Worksheet: > with(Student[Calculus1]); > Student[Calculus1][ LimitTutor]();

Fig. 16. Enter the function 9

Fig. 17. All steps of the solution, part 1

Fig. 18. All steps of the solution, part 2

Step-by-step Solution: > with(Student[Calculus1]); > ShowSolution(Limit((1+cot(x))^tan(x), x = (1/2)*Pi));

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Solution by using standard commands: > with(Student[Calculus1]); > limit((1+cot(x))^tan(x), x = (1/2)*Pi); 11

4. CONCLUSION Maple helps students to understand the algorithm of a solution of a math problem. Some interactive opportunities are related to the usage of Tutors, Assistants or ShowSolution command. In the future, we will extend the research topics to other calculus and engineering mathematics’ problems and solve these problems using Maple. These results will be used as teaching materials for Maple for education and research in order to enhance the connotations of calculus and engineering mathematics. 5. REFERENCES Alharbi A., F. Tcheir, and M. Siddique. (2016) A Mathematics E-book Application by Maple Animations. Int'l Conf. Frontiers in Education: CS and CE (FECS'16 ) Kumar, A., & Kumaresan, S. (2008). Use of mathematical software for teaching and learning mathematics. In Proceedings of ICME 2008 Salleh T. S, Zakaria E. (2016) The Effects of Maple Integrated Strategy on Engineering Technology Students’ Understanding of Integral Calculus. The Turkish Online Journal of Educational Technology – July 2016, volume 15 issue 3 Tonchev, J. (2013) Maple - Transformations, Calculations, Vizualization. Publishing house Technics, Sofia, 240 p. Yu Chii-Huei. (2016) Expressions of Some Complicated Integrals. International Journal of Scientific Research in Science and Technology, Themed Section: Science and Technology, Volume 2, Issue 1, Online ISSN: 2395-602X, pp. 59-62 Zakaria E., Salleh T. S. (2015) Using Technology in Learning Integral Calculus. Mediterranean Journal of Social Sciences. MCSER Publishing, Rome-Italy, Vol 6 No 5 S1, pp. 144-148 Maple (May 8, 2017) https://www.maplesoft.com/ CompMath (May 8, 2017) http://www.CompMath.eu/ FutureMath project (May 8, 2017) http://civile.utcb.ro/FutureMath/

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